Let $z$ and $w$ be two complex numbers such that $w = z \bar { z } - 2 z + 2 , \left| \frac { z + i } { z - 3 i } \right| = 1$ and $\operatorname { Re } ( w )$ has minimum value. Then, the minimum value of $n \in N$ for which $w ^ { n }$ is real, is equal to $\_\_\_\_$.